Assessment and Future Directions of Nonlinear Model Predictive ...

MPC for Stochastic Systems 263( )κ 2 cT 1/22 Z 2 (k + j|k)c 2 ≤ Y2 − c T 2 ¯x(k + j|k) (23)where κ 2 satisfies N(κ 2 )=p 2 . Similarly, making use of confidence ellipsoids forq(k), the terminal constraint (22d) can be expressedr 1(vTi Z 2 (k + j|k)v i) 1/2≤ 1 − vTi ¯x(k + j|k), i =1,...,m (24)where r 1 is defined by Pr ( χ 2 (NL) ≤ r1) 2 = p2 . It follows that (22) is convex,and has the form of a second-order cone program (SOCP), enabling solution viaefficient algorithms [17]. The stability properties of the MPC law can be statedas follows.Theorem 2. Assume that (22) is feasible at all times k =0, 1,....Theny 1 (k) →0,and‖x(k)‖ 2 converges to a finite limit with probability 1 if (A, c 1 ) is observable.Proof. From Lemmas 1 and 3, the cost, ˜J k+1 , for the suboptimal sequence ũ(k +1) = {u ∗ (k+1|k),...,Kx ∗ (k+N|k)} at time k+1 satisfies E k ˜Jk+1 ≤ Jk ∗ −y2 1 (k),where Jk ∗ is the optimal value of (22a). After optimization at k +1wehaveE k Jk+1 ∗ ≤ E k ˜J k+1 − y1 2 (k) ≤ J k ∗ − y2 1 (k) (25)It follows that J k converges to a lower limit and y 1 (k) → 0 with probability1 [18]. Furthermore the definitions of stage cost (5) and terminal penalty (17)imply that∞∑J k = c T 1 ¯x(k + j|k)¯xT (k + j|k)c 1 + κ 2 1 cT 1 Z 2(k + j|k)c 1j=0and, since ∑ ∞j=0 cT 1 Z 2 (k+j|k)c 1 is positive definite in u(k)if(A, c 1 ) is observable,it follows that J k is positive definite in x(k) if(A, c 1 ) is observable. Under thiscondition therefore, ‖x(k)‖ 2 converges to a finite limit with probability 1.Note that the derivation of (25) assumes a pre-stabilized prediction model;the same convergence property can otherwise be ensured by using a variablehorizon N.The constraints (22b-d) apply only to predicted trajectories at time k, anddo not ensure feasibility of (22) at future times. For example, at time k +1, (23)requiresκ 2(cT2 Z 2 (k + j|k +1)c 2) 1/2≤ Y2 − c T 2 ¯x(k + j|k +1),j =1,...,Nwhere ¯x(k+j|k+1) is a Gaussian random variable at time k,withmean¯x(k+j|k)and variance c T 2 Aj−1 Z 2 (k +1|k)A j−1T c 2 . Therefore (23) is feasible at k +1 withprobability p 2 ifκ 2(cT2 Z 2 (k+j|k+1)c 2) 1/2+κ2 (cT2 A j−1 Z 2 (k+1|k)A j−1T c 2) 1/2≤ Y2 −c T 2 ¯x(k+j|k)holds for j =1,...,N at time k; this condition is necessarily more restrictivethan (23) since Z 2 (k + j|k) =Z 2 (k + j|k +1)+A j−1 Z 2 (k +1|k)A j−1T .Inorderto provide a recursive guarantee of feasibility we therefore include additionalconstraints in the online optimization, as summarized in the following result.